Mathematics for the interested outsider

Projecting Onto Invariants

Given a -module , we can find the -submodule of -invariant vectors. It’s not just a submodule, but it’s a direct summand. Thus not only does it come with an inclusion mapping , but there must be a projection . That is, there’s a linear map that takes a vector and returns a -invariant vector, and further if the vector is already -invariant it is left alone.

Well, we know that it exists, but it turns out that we can describe it rather explicitly. The projection from vectors to -invariant vectors is exactly the “averaging” procedure we ran into (with a slight variation) when proving Maschke’s theorem. We’ll describe it in general, and then come back to see how it applies in that case.

Given a vector , we define

This is clearly a linear operation. I say that is invariant under the action of . Indeed, given we calculate

since as ranges over , so does , albeit in a different order. Further, if is already -invariant, then we find

so this is indeed the projection we’re looking for.

Now, how does this apply to Maschke’s theorem? Well, given a -module , the collection of sesquilinear forms on the underlying space forms a vector space itself. Indeed, such forms correspond to correspond to Hermitian matrices, which form a vector space. Anyway, rather than write the usual angle-brackets, we will write one of these forms as a bilinear function .

Now I say that the space of forms carries an action from the right by . Indeed, we can define

It’s straightforward to verify that this is a right action by . So, how do we “average” the form to get a -invariant form? We define

which — other than the factor of — is exactly how we came up with a -invariant form in the proof of Maschke’s theorem!

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[…] can get a more explicit description to verify this equivalence by projecting onto the invariants. Given a tensor , we consider it instead as a tensor in . Now, this is far from unique, since many […]

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